Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-x-5y &= 3 \\ 2x+8y &= -7\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $1$ $\begin{align*}-2x-10y &= 6\\ 2x+8y &= -7\end{align*}$ Add the top and bottom equations. $-2y = -1$ Divide both sides by $-2$ and reduce as necessary. $y = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $y$ in the top equation. $-x-5( \dfrac{1}{2}) = 3$ $-x-\dfrac{5}{2} = 3$ $-x = \dfrac{11}{2}$ $x = -\dfrac{11}{2}$ The solution is $\enspace x = -\dfrac{11}{2}, \enspace y = \dfrac{1}{2}$.